Abstract group 216.110: $C_6^2:C_6$

• Label: 216.110
• Order: \( 2^{3} \cdot 3^{3} \)
• Exponent: \( 2 \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 3 \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{5} \cdot 3^{4} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \cdot 3 \)
• Perm deg.: $13$
• Trans deg.: $36$
• Rank: $3$

Group information

Description:	$C_6^2:C_6$
Order:	\(216\)\(\medspace = 2^{3} \cdot 3^{3} \)
Exponent:	\(6\)\(\medspace = 2 \cdot 3 \)
Automorphism group:	$C_3^2:D_6\times S_4$, of order \(2592\)\(\medspace = 2^{5} \cdot 3^{4} \)
Composition factors:	$C_2$ x 3, $C_3$ x 3
Derived length:	$2$

This group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.

Group statistics

Order	1	2	3	6
Elements	1	39	26	150	216
Conjugacy classes	1	7	6	26	40
Divisions	1	7	4	16	28
Autjugacy classes	1	2	4	5	12

Dimension	1	2	4	6
Irr. complex chars.	24	12	0	4	40
Irr. rational chars.	8	12	4	4	28

Minimal presentations

Permutation degree:	$13$
Transitive degree:	$36$
Rank:	$3$
Inequivalent generating triples:	$1092$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	7	7	7

Constructions

Presentation:	$\langle a, b, c \mid a^{6}=b^{6}=c^{6}=[b,c]=1, b^{a}=b^{5}c^{2}, c^{a}=c^{5} \rangle$
Permutation group:	Degree $13$ $\langle(3,7)(5,6)(8,9)(10,11)(12,13), (10,12)(11,13), (10,13)(11,12), (1,2,4)(3,6,8)(5,9,7), (2,5,6)(4,8,9), (1,3,7)(2,6,5)(4,8,9)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 2 & 0 & 2 & 0 \\ 1 & 0 & 2 & 1 \end{array}\right), \left(\begin{array}{rrrr} 1 & 1 & 2 & 2 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 2 & 1 & 2 \end{array}\right), \left(\begin{array}{rrrr} 2 & 2 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 1 & 2 & 1 & 1 \\ 2 & 1 & 0 & 0 \end{array}\right), \left(\begin{array}{rrrr} 2 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 2 \end{array}\right), \left(\begin{array}{rrrr} 0 & 2 & 0 & 1 \\ 0 & 1 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 2 & 2 & 2 & 0 \end{array}\right), \left(\begin{array}{rrrr} 2 & 1 & 2 & 2 \\ 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 2 \\ 1 & 0 & 1 & 1 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{3})$
Transitive group:	36T204	36T208			more information
Direct product:	$C_2$ ${}^2$ $\, \times\, $ $(C_3^2:C_6)$
Semidirect product:	$C_6^2$ $\,\rtimes\,$ $S_3$	$C_6^2$ $\,\rtimes\,$ $C_6$	$\He_3$ $\,\rtimes\,$ $C_2^3$	$(C_6:S_3)$ $\,\rtimes\,$ $C_6$ (6)	all 12
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_6$ . $(C_6\times S_3)$ (3)	$C_3$ . $(C_6\times D_6)$	$(C_2\times C_6)$ . $(C_3\times S_3)$		more information

Elements of the group are displayed as words in the presentation generators from the presentation above.

Homology

Abelianization:	$C_{2}^{2} \times C_{6} \simeq C_{2}^{3} \times C_{3}$
Schur multiplier:	$C_{2}^{2} \times C_{6}$
Commutator length:	$1$

Subgroups

There are 480 subgroups in 122 conjugacy classes, 47 normal (12 characteristic).

Characteristic subgroups are shown in this color. Normal (but not characteristic) subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $C_3^2:C_6$
Commutator:	$G' \simeq$ $C_3^2$	$G/G' \simeq$ $C_2^2\times C_6$
Frattini:	$\Phi \simeq$ $C_3$	$G/\Phi \simeq$ $C_6\times D_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^2\times \He_3$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_6^2:C_6$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_6$	$G/\operatorname{soc} \simeq$ $C_3\times S_3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^3$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $\He_3$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

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Normal subgroups

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 216: $C_6^2:C_6$

Order 108: $C_3^2:D_6$ x 6, $C_2^2\times \He_3$

Order 72: $C_6:D_6$, $C_6\times D_6$

Order 54: $C_3^2:C_6$ x 4, $C_2\times \He_3$ x 3

Order 36: $C_6:S_3$ x 6, $C_6\times S_3$ x 6, $C_6^2$ x 3

Order 27: $\He_3$

Order 24: $C_2\times D_6$ x 2, $C_2^2\times C_6$

Order 18: $C_3\times C_6$ x 9, $C_3:S_3$ x 4, $C_3\times S_3$ x 4

Order 12: $D_6$ x 12, $C_2\times C_6$ x 10

Order 9: $C_3^2$ x 3

Order 8: $C_2^3$

Order 6: $C_6$ x 16, $S_3$ x 8

Order 4: $C_2^2$ x 7

Order 3: $C_3$ x 4

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 216: $C_6^2:C_6$

Order 108: $C_2^2\times \He_3$, $C_3^2:D_6$

Order 72: $C_6:D_6$, $C_6\times D_6$

Order 54: $C_3^2:C_6$, $C_2\times \He_3$

Order 36: $C_6^2$ x 3, $C_6:S_3$, $C_6\times S_3$

Order 27: $\He_3$

Order 24: $C_2\times D_6$ x 2, $C_2^2\times C_6$

Order 18: $C_3\times C_6$ x 3, $C_3:S_3$, $C_3\times S_3$

Order 12: $C_2\times C_6$ x 5, $D_6$ x 2

Order 9: $C_3^2$ x 3

Order 8: $C_2^3$

Order 6: $C_6$ x 5, $S_3$ x 2

Order 4: $C_2^2$ x 2

Order 3: $C_3$ x 4

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 216: $C_6^2:C_6$ ($C_1$)

Order 108: $C_3^2:D_6$ ($C_2$) x 6, $C_2^2\times \He_3$ ($C_2$)

Order 72: $C_6:D_6$ ($C_3$)

Order 54: $C_3^2:C_6$ ($C_2^2$) x 4, $C_2\times \He_3$ ($C_2^2$) x 3

Order 36: $C_6:S_3$ ($C_6$) x 6, $C_6^2$ ($C_6$), $C_6^2$ ($S_3$)

Order 27: $\He_3$ ($C_2^3$)

Order 18: $C_3:S_3$ ($C_2\times C_6$) x 4, $C_3\times C_6$ ($C_2\times C_6$) x 3, $C_3\times C_6$ ($D_6$) x 3

Order 12: $C_2\times C_6$ ($C_3\times S_3$)

Order 9: $C_3^2$ ($C_2^2\times C_6$), $C_3^2$ ($C_2\times D_6$)

Order 6: $C_6$ ($C_6\times S_3$) x 3

Order 4: $C_2^2$ ($C_3^2:C_6$)

Order 3: $C_3$ ($C_6\times D_6$)

Order 2: $C_2$ ($C_3^2:D_6$) x 3

Order 1: $C_1$ ($C_6^2:C_6$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 216: $C_6^2:C_6$ ($C_1$)

Order 108: $C_2^2\times \He_3$ ($C_2$), $C_3^2:D_6$ ($C_2$)

Order 72: $C_6:D_6$ ($C_3$)

Order 54: $C_3^2:C_6$ ($C_2^2$), $C_2\times \He_3$ ($C_2^2$)

Order 36: $C_6^2$ ($C_6$), $C_6^2$ ($S_3$), $C_6:S_3$ ($C_6$)

Order 27: $\He_3$ ($C_2^3$)

Order 18: $C_3\times C_6$ ($C_2\times C_6$), $C_3\times C_6$ ($D_6$), $C_3:S_3$ ($C_2\times C_6$)

Order 12: $C_2\times C_6$ ($C_3\times S_3$)

Order 9: $C_3^2$ ($C_2^2\times C_6$), $C_3^2$ ($C_2\times D_6$)

Order 6: $C_6$ ($C_6\times S_3$)

Order 4: $C_2^2$ ($C_3^2:C_6$)

Order 3: $C_3$ ($C_6\times D_6$)

Order 2: $C_2$ ($C_3^2:D_6$)

Order 1: $C_1$ ($C_6^2:C_6$)

Series

Derived series

$C_6^2:C_6$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

Chief series

$C_6^2:C_6$

$\rhd$

$C_2^2\times \He_3$

$\rhd$

$C_6^2$

$\rhd$

$C_3\times C_6$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_6^2:C_6$

$\rhd$

$C_3^2$

Upper central series

$C_1$

$\lhd$

$C_2^2$

Supergroups

This group is a maximal subgroup of 38 larger groups in the database.

This group is a maximal quotient of 34 larger groups in the database.

Character theory

Complex character table

See the $40 \times 40$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $28 \times 28$ rational character table.